Let us set some global options for all code chunks in this document.

# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = TRUE,    
  # Disable warnings printed by R code chunks
  warning = TRUE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}
# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  floor(x * 100) / 100
}
# install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
## Loading required package: Matrix
## This is INLA_25.04.16 built 2025-04-16 08:05:23 UTC.
##  - See www.r-inla.org/contact-us for how to get help.
##  - List available models/likelihoods/etc with inla.list.models()
##  - Use inla.doc(<NAME>) to access documentation
library(inlabru)
## Loading required package: fmesher
library(rSPDE)
## This is rSPDE 2.5.1
## - See https://davidbolin.github.io/rSPDE for vignettes and manuals.
library(MetricGraph)
## This is MetricGraph 1.4.1
## - See https://davidbolin.github.io/MetricGraph for vignettes and manuals.
## 
## Attaching package: 'MetricGraph'
## The following object is masked from 'package:stats':
## 
##     filter
library(grateful)

library(plotly)
## Loading required package: ggplot2
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout

We want to solve the fractional diffusion equation \[\begin{equation} \label{eq:maineq} \partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma, \end{equation}\] where \(u\) satisfies the Kirchhoff vertex conditions \[\begin{equation} \label{eq:Kcond} \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\} \end{equation}\]

If \(f=0\), then the solution is given by \[\begin{equation} \label{eq:sol_reprentation} u(s,t) = \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\frac{\alpha}{2}}_jt}\left(u_0, e_j\right)_{L_2(\Gamma)}e_j(s). \end{equation}\]

# Function to build a tadpole graph and create a mesh
gets_graph_tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 100)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(h = h)
  return(graph)
}

Let \(\Gamma_T = (\mathcal{V},\mathcal{E})\) characterize the tadpole graph with \(\mathcal{V}= \{v_1,v_2\}\) and \(\mathcal{E}= \{e_1,e_2\}\) as specified in Figure \(\ref{Interval.Circle.Tadpole}\)c. The left edge \(e_1\) has length 1 and the circular edge \(e_2\) has length 2. As discussed in Subsection \(\ref{subsec:prelim}\), a point on \(e_1\) is parameterized via \(s=\left(e_1, t\right)\) for \(t \in[0,1]\) and a point on \(e_2\) via \(s=\left(e_2, t\right)\) for \(t\in[0,2]\). One can verify that \(-\Delta_\Gamma\) has eigenvalues \(0,\left\{(i \pi / 2)^2\right\}_{i \in \mathbb{N}}\) and \(\left\{(i \pi / 2)^2\right\}_{2 i \in \mathbb{N}}\) with corresponding eigenfunctions \(\phi_0\), \(\left\{\phi_i\right\}_{i \in \mathbb{N}}\), and \(\left\{\psi_i\right\}_{2 i \in \mathbb{N}}\) given by \(\phi_0(s)=1 / \sqrt{3}\) and \[\begin{equation*} \phi_i(s)=C_{\phi, i}\begin{cases} -2 \sin (\frac{i\pi}{2}) \cos (\frac{i \pi t}{2}), & s \in e_1, \\ \sin (i \pi t / 2), & s \in e_2, \end{cases}, \quad \psi_i(s)=\frac{\sqrt{3}}{\sqrt{2}} \begin{cases} (-1)^{i / 2} \cos (\frac{i \pi t}{2}), & s \in e_1, \\ \cos (\frac{i \pi t}{2}), & s \in e_2, \end{cases}, \end{equation*}\] where \(C_{\phi, i}=1\) if \(i\) is even and \(C_{\phi, i}=1 / \sqrt{3}\) otherwise. Moreover, these functions form an orthonormal basis for \(L_2(\Gamma_T)\).

# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}

return(f)
}

Implementation of \(u\)

h <- 0.01
graph <- gets_graph_tadpole(h = h)
## Starting graph creation...
## LongLat is set to FALSE
## Creating edges...
## Setting edge weights...
## Computing bounding box...
## Setting up edges
## Merging close vertices
## Total construction time: 0.15 secs
## Creating and updating vertices...
## Storing the initial graph...
## Computing the relative positions of the edges...
T_final <- 2
time_step <- 0.01
time_seq <- seq(0, T_final, by = time_step)
# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
I <- Matrix::Diagonal(nrow(C))
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
edge_number <- graph$mesh$VtE[, 1]
pos <- sum(edge_number == 1)+1
order_to_plot <- function(v)return(c(v[1], v[3:pos], v[2], v[(pos+1):length(v)], v[2]))
weights <- graph$mesh$weights
# Initial condition
U_0 <- 10*exp(-((x-1)^2 + (y)^2))
kappa <- 1
alpha <- 0.57
beta <- alpha/2
L <- kappa^2*C + G
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
Ci <- op$Ci
n_finite <- 1000

U_true <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_true[, 1] <- U_0
for (k in 1:(length(time_seq) - 1)) {
  aux_k <- rep(0, nrow(C))
  for (j in 0:n_finite) {
    decay_j <- exp(-time_seq[k+1]*(kappa^2 + (j*pi/2)^2)^(alpha/2))
    e_j <- tadpole.eig(j,graph)$phi
    aux_k <- aux_k + decay_j*sum(U_0*e_j*weights)*e_j
    if (j>0 && (j %% 2) == 0) {
      e_j <- tadpole.eig(j,graph)$psi
      aux_k <- aux_k + decay_j*sum(U_0*e_j*weights)*e_j
      }
    }
  U_true[, k + 1] <- aux_k
}
Pr.apply.mult <- function(v){return(Pr.mult(op, v))}
Pl.apply.solve <- function(v){return(Pl.solve(op, v))}
PliC <- apply(C, 2, Pl.apply.solve) # PlC^-1
# Precompute the LHS1 matrix
aux <- apply(PliC, 2, Pr.apply.mult) #PrPl^-1C
LHS <- aux + time_step * Matrix::Diagonal(nrow(C)) 

# Initialize U matrix to store solution at each time step
U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- aux %*% U_approx[, k]
  U_approx[, k + 1] <- as.matrix(solve(LHS, RHS))
}
x <- order_to_plot(x)
y <- order_to_plot(y)
max_error_at_each_time <- apply(abs(U_true - U_approx), 2, max)

U_true <- apply(U_true, 2, order_to_plot)
U_approx <- apply(U_approx, 2, order_to_plot)



# Create interactive plot
fig <- plot_ly()


# Add second line (max_error_at_each_time)
fig <- fig %>% add_trace(
  x = ~time_seq, y = ~max_error_at_each_time, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'red', width = 2),
  marker = list(size = 4),
  name = "Max Error True and Approx 2"
)

# Layout
fig <- fig %>% layout(
  title = "Max Error at Each Time Step",
  xaxis = list(title = "Time"),
  yaxis = list(title = "Max Error"),
  legend = list(x = 0.1, y = 0.9)
)


plot_data <- data.frame(
  x = rep(x, times = ncol(U_true)),
  y = rep(y, times = ncol(U_true)),
  z_true = as.vector(U_true),
  z_approx = as.vector(U_approx),
  frame = rep(time_seq, each = length(x))
)

# Compute axis limits
x_range <- range(x)
y_range <- range(y)
z_range <- range(c(U_true, U_approx))

# Initial plot setup (first frame only)
p <- plot_ly(plot_data, frame = ~frame) %>%
  add_trace(
    x = ~x, y = ~y, z = ~z_true,
    type = "scatter3d", mode = "lines",
    name = "True",
    line = list(color = "blue", width = 2)
  ) %>%
  add_trace(
    x = ~x, y = ~y, z = ~z_approx,
    type = "scatter3d", mode = "lines",
    name = "Approx",
    line = list(color = "red", width = 2)
  ) %>%
  layout(
    scene = list(
      xaxis = list(title = "x", range = x_range),
      yaxis = list(title = "y", range = y_range),
      zaxis = list(title = "Value", range = z_range)
    ),
    updatemenus = list(
      list(
        type = "buttons", showactive = FALSE,
        buttons = list(
          list(label = "Play", method = "animate",
               args = list(NULL, list(frame = list(duration = 100, redraw = TRUE), fromcurrent = TRUE))),
          list(label = "Pause", method = "animate",
               args = list(NULL, list(mode = "immediate", frame = list(duration = 0), redraw = FALSE)))
        )
      )
    ),
    title = "Time: 0"
  )

# Convert to plotly object with frame info
pb <- plotly_build(p)

# Inject custom titles into each frame
for (i in seq_along(pb$x$frames)) {
  t <- time_seq[i]
  err <- signif(max_error_at_each_time[i], 4)
  pb$x$frames[[i]]$layout <- list(title = paste0("Time: ", t, " | Max Error: ", err))
}

# Create 3D scatter plot
pic <- plot_ly(x = ~x, y = ~y, z = ~U_true[,2], type = 'scatter3d', mode = 'lines')
fig  # Display the plot
pb

Figure 1: Caption

pic
---
title: "Solving a parabolic equation"
date: "Created: 20-04-2025. Last modified: `r format(Sys.time(), '%d-%m-%Y.')`"
output:
  html_document:
    mathjax: "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
    highlight: pygments
    theme: flatly
    code_folding: show # class.source = "fold-hide" to hide code and add a button to show it
    df_print: paged
    toc: true
    toc_float:
      collapsed: true
      smooth_scroll: true
    number_sections: false
    fig_caption: true
    code_download: true
always_allow_html: true
bibliography: 
  - references.bib
  - grateful-refs.bib
header-includes:
  - \newcommand{\ar}{\mathbb{R}}
  - \newcommand{\llav}[1]{\left\{#1\right\}}
  - \newcommand{\pare}[1]{\left(#1\right)}
  - \newcommand{\Ncal}{\mathcal{N}}
  - \newcommand{\Vcal}{\mathcal{V}}
  - \newcommand{\Ecal}{\mathcal{E}}
  - \newcommand{\Wcal}{\mathcal{W}}
---

```{r xaringanExtra-clipboard, echo = FALSE}
htmltools::tagList(
  xaringanExtra::use_clipboard(
    button_text = "<i class=\"fa-solid fa-clipboard\" style=\"color: #00008B\"></i>",
    success_text = "<i class=\"fa fa-check\" style=\"color: #90BE6D\"></i>",
    error_text = "<i class=\"fa fa-times-circle\" style=\"color: #F94144\"></i>"
  ),
  rmarkdown::html_dependency_font_awesome()
)
```


```{css, echo = FALSE}
body .main-container {
  max-width: 100% !important;
  width: 100% !important;
}
body {
  max-width: 100% !important;
}

body, td {
   font-size: 16px;
}
code.r{
  font-size: 14px;
}
pre {
  font-size: 14px
}
.custom-box {
  background-color: #f5f7fa; /* Light grey-blue background */
  border-color: #e1e8ed; /* Light border color */
  color: #2c3e50; /* Dark text color */
  padding: 15px; /* Padding inside the box */
  border-radius: 5px; /* Rounded corners */
  margin-bottom: 20px; /* Spacing below the box */
}
.caption {
  margin: auto;
  text-align: center;
  margin-bottom: 20px; /* Spacing below the box */
}
```


Let us set some global options for all code chunks in this document.


```{r}
# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = TRUE,    
  # Disable warnings printed by R code chunks
  warning = TRUE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}
# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  floor(x * 100) / 100
}
```




```{r}
# install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(grateful)

library(plotly)
```


We want to solve the fractional diffusion equation
\begin{equation}
\label{eq:maineq}
    \partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma,
\end{equation}
where $u$ satisfies the Kirchhoff vertex conditions
\begin{equation}
\label{eq:Kcond}
    \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\}
\end{equation}

If $f=0$, then the solution is given by
\begin{equation}
\label{eq:sol_reprentation}
        u(s,t) = \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\frac{\alpha}{2}}_jt}\left(u_0, e_j\right)_{L_2(\Gamma)}e_j(s).
\end{equation}

```{r}
# Function to build a tadpole graph and create a mesh
gets_graph_tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 100)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges = list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$build_mesh(h = h)
  return(graph)
}
```

Let $\Gamma_T = (\Vcal,\Ecal)$ characterize the tadpole graph with $\Vcal = \{v_1,v_2\}$ and $\Ecal = \{e_1,e_2\}$ as specified in Figure \ref{Interval.Circle.Tadpole}c. The left edge $e_1$ has length 1 and the circular edge $e_2$ has length 2. As discussed in Subsection \ref{subsec:prelim}, a point on $e_1$ is parameterized via $s=\left(e_1, t\right)$ for $t \in[0,1]$ and a point on $e_2$ via $s=\left(e_2, t\right)$ for $t\in[0,2]$. One can verify that $-\Delta_\Gamma$ has eigenvalues $0,\left\{(i \pi / 2)^2\right\}_{i \in \mathbb{N}}$ and $\left\{(i \pi / 2)^2\right\}_{2 i \in \mathbb{N}}$ with corresponding eigenfunctions $\phi_0$, $\left\{\phi_i\right\}_{i \in \mathbb{N}}$, and $\left\{\psi_i\right\}_{2 i \in \mathbb{N}}$ given by $\phi_0(s)=1 / \sqrt{3}$ and 
\begin{equation*}
    \phi_i(s)=C_{\phi, i}\begin{cases}
        -2 \sin (\frac{i\pi}{2}) \cos (\frac{i \pi t}{2}), & s \in e_1, \\
\sin (i \pi t / 2), & s \in e_2,
    \end{cases},
\quad 
    \psi_i(s)=\frac{\sqrt{3}}{\sqrt{2}} \begin{cases}
    (-1)^{i / 2} \cos (\frac{i \pi t}{2}), & s \in e_1, \\
\cos (\frac{i \pi t}{2}), & s \in e_2,
\end{cases},
\end{equation*}
where $C_{\phi, i}=1$ if $i$ is even and $C_{\phi, i}=1 / \sqrt{3}$ otherwise. Moreover, these functions form an orthonormal basis for $L_2(\Gamma_T)$.

```{r}
# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}

return(f)
}
```

Implementation of $u$

```{r}
h <- 0.01
graph <- gets_graph_tadpole(h = h)
T_final <- 2
time_step <- 0.01
time_seq <- seq(0, T_final, by = time_step)
# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
I <- Matrix::Diagonal(nrow(C))
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
edge_number <- graph$mesh$VtE[, 1]
pos <- sum(edge_number == 1)+1
order_to_plot <- function(v)return(c(v[1], v[3:pos], v[2], v[(pos+1):length(v)], v[2]))
weights <- graph$mesh$weights
# Initial condition
U_0 <- 10*exp(-((x-1)^2 + (y)^2))
```


```{r}
kappa <- 1
alpha <- 0.57
beta <- alpha/2
L <- kappa^2*C + G
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
Ci <- op$Ci
```



```{r}
n_finite <- 1000

U_true <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_true[, 1] <- U_0
for (k in 1:(length(time_seq) - 1)) {
  aux_k <- rep(0, nrow(C))
  for (j in 0:n_finite) {
    decay_j <- exp(-time_seq[k+1]*(kappa^2 + (j*pi/2)^2)^(alpha/2))
    e_j <- tadpole.eig(j,graph)$phi
    aux_k <- aux_k + decay_j*sum(U_0*e_j*weights)*e_j
    if (j>0 && (j %% 2) == 0) {
      e_j <- tadpole.eig(j,graph)$psi
      aux_k <- aux_k + decay_j*sum(U_0*e_j*weights)*e_j
      }
    }
  U_true[, k + 1] <- aux_k
}
```

```{r}

Pr.apply.mult <- function(v){return(Pr.mult(op, v))}
Pl.apply.solve <- function(v){return(Pl.solve(op, v))}
PliC <- apply(C, 2, Pl.apply.solve) # PlC^-1
# Precompute the LHS1 matrix
aux <- apply(PliC, 2, Pr.apply.mult) #PrPl^-1C
LHS <- aux + time_step * Matrix::Diagonal(nrow(C)) 

# Initialize U matrix to store solution at each time step
U_approx <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- aux %*% U_approx[, k]
  U_approx[, k + 1] <- as.matrix(solve(LHS, RHS))
}
```



```{r}
x <- order_to_plot(x)
y <- order_to_plot(y)
max_error_at_each_time <- apply(abs(U_true - U_approx), 2, max)

U_true <- apply(U_true, 2, order_to_plot)
U_approx <- apply(U_approx, 2, order_to_plot)



# Create interactive plot
fig <- plot_ly()


# Add second line (max_error_at_each_time)
fig <- fig %>% add_trace(
  x = ~time_seq, y = ~max_error_at_each_time, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'red', width = 2),
  marker = list(size = 4),
  name = "Max Error True and Approx 2"
)

# Layout
fig <- fig %>% layout(
  title = "Max Error at Each Time Step",
  xaxis = list(title = "Time"),
  yaxis = list(title = "Max Error"),
  legend = list(x = 0.1, y = 0.9)
)


plot_data <- data.frame(
  x = rep(x, times = ncol(U_true)),
  y = rep(y, times = ncol(U_true)),
  z_true = as.vector(U_true),
  z_approx = as.vector(U_approx),
  frame = rep(time_seq, each = length(x))
)

# Compute axis limits
x_range <- range(x)
y_range <- range(y)
z_range <- range(c(U_true, U_approx))

# Initial plot setup (first frame only)
p <- plot_ly(plot_data, frame = ~frame) %>%
  add_trace(
    x = ~x, y = ~y, z = ~z_true,
    type = "scatter3d", mode = "lines",
    name = "True",
    line = list(color = "blue", width = 2)
  ) %>%
  add_trace(
    x = ~x, y = ~y, z = ~z_approx,
    type = "scatter3d", mode = "lines",
    name = "Approx",
    line = list(color = "red", width = 2)
  ) %>%
  layout(
    scene = list(
      xaxis = list(title = "x", range = x_range),
      yaxis = list(title = "y", range = y_range),
      zaxis = list(title = "Value", range = z_range)
    ),
    updatemenus = list(
      list(
        type = "buttons", showactive = FALSE,
        buttons = list(
          list(label = "Play", method = "animate",
               args = list(NULL, list(frame = list(duration = 100, redraw = TRUE), fromcurrent = TRUE))),
          list(label = "Pause", method = "animate",
               args = list(NULL, list(mode = "immediate", frame = list(duration = 0), redraw = FALSE)))
        )
      )
    ),
    title = "Time: 0"
  )

# Convert to plotly object with frame info
pb <- plotly_build(p)

# Inject custom titles into each frame
for (i in seq_along(pb$x$frames)) {
  t <- time_seq[i]
  err <- signif(max_error_at_each_time[i], 4)
  pb$x$frames[[i]]$layout <- list(title = paste0("Time: ", t, " | Max Error: ", err))
}

# Create 3D scatter plot
pic <- plot_ly(x = ~x, y = ~y, z = ~U_true[,2], type = 'scatter3d', mode = 'lines')
```


```{r}
fig  # Display the plot
```


```{r, fig.height = 6, out.width = "100%", fig.cap = captioner("Caption")}
pb
```

```{r}
pic
```


